3 Answers
3

There is a subtle difference; maximum and minimum relate to absolute values — there is nothing higher than the maximum and nothing lower than the minimum. Maximal and minimal, however, can be more vague.

In "I want to buy this at minimal cost" and "this action carries a minimal risk", minimal means "very small" as opposed to "the lowest possible"; the same distinction is true of maximum and maximal.

I don't believe this answer is correct. The usage you cite is simply a misunderstanding about 'maximum' etc., which are frequently used to mean 'very big' rather than their correct meaning which is 'the biggest'.
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DJClayworthAug 17 '11 at 14:29

Surely if words are frequently used to mean something then that becomes their correct meaning? Language isn't static. After all the question was about difference "in usage" not "in original meaning"
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WaggersAug 18 '11 at 7:53

Waggers' answer does an excellent job of explaining the difference in the non-technical meaning. In some areas of mathematics (e.g., in maximal element, maximal matching), a maximal value (you can't use the with this meaning) is essentially a local maximum—it's a maximum value in its neighborhood; i.e., there are no small changes which will increase the value. See this Wikipedia article for a more technical description in relation to partially ordered sets.

It seems to me that mathematical definition (a maximal value=the local maximum) effectively means the same as @Waggers "non-technical" one anyway. A maximal value would not be so called if it was easy to identify a higher one "nearby" (whatever "nearby" meant in context).
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FumbleFingersAug 17 '11 at 16:39

The short answer is that, unless you are a mathematician or an economist, there is no difference. However, there is a fine distinction between the two terms that can be relevant in the context of non linearly ordered sets (i.e. sets in which not every pair of elements need be comparable).

An element is maximal if there is no other element greater.

An element is maximum if it is itself greater than every other element.

If the "elements" under discussion are numbers, the definitions coincide, but there are contexts in which the distinction matters.

For example, in an election one might say that candidate 1 is strictly better than candidate 2 if all potential voters prefer candidate 1 to candidate 2. Say that three candidates - Mitt, Barry, and Adolf - are running for president of a club.

The club members are divided into two contingents of equal size. One group unanimously prefers Barry to Mitt and Mitt to Adolf, while the other unanimously prefers Mitt to Barry and Barry to Adolf.

Barry and Mitt are both strictly better candidates than Adolf, as all members rank Adolf last. But since some members prefer Barry to Mitt and some prefer Mitt to Barry, neither is strictly better than the other. Thus neither can be maximum with respect to this ordering. However, because no candidates exist that are strictly better than either of them, both candidates are maximal with respect to this ordering.